Materials exhibiting different properties along different locations within their body are called heterogeneous. Their heterogeneity may be random or organized. Layered media are examples of materials with organized heterogeneity that exhibit similar properties within a bed plane and different properties perpendicular to the bed plane. These types of materials may be modeled by assuming rotational symmetry in material properties, with an axis of rotation perpendicular to bedding. Because of the laminated structure (which may be fine-scale, as in shales, or large-scale, as in reservoir interbeds) their stress-strain relationships change with orientation to bedding. In general, laminated materials tend to be stiffer along the direction parallel to bedding and more compliant along the direction perpendicular to bedding. Correspondingly, propagating sound waves (compressional and shear) in these materials result in wave velocities that are higher parallel to bedding and lower perpendicular to bedding. The theory of elastic anisotropy describes this behavior. By defining material properties along principal directions of material symmetry, it provides a methodology for predicting material behavior under any conditions of applied loading and deformation.
Geologic materials are complex and often exhibit various types of heterogeneity (e.g., fine-scale texture superposed to the presence of fracture sets and as part of a larger scale structure). Furthermore, the layering may not be ideal (e.g., some beds may have different orientations or be discontinuous). The resulting stress-strain behavior may or may not be well represented by the elastic anisotropic theory, and may change with scale (from sample-scale to log-scale). Furthermore, their stress-strain behavior may not be elastic (e.g., plastic shales).
There are three basic types of formations: formations that have identical elastic properties in all three spatial directions are called isotropic, formations that have identical elastic properties in two directions but a different property in the third direction are called transverse isotropic (TI), and formations that have different properties in all three dimensions are called orthotropic. There are two independent moduli that characterize isotropic formations, five independent moduli that characterize TI formations and nine independent moduli that characterize orthorhombic formations. The aforementioned types of isotropic formations may be modeled using isotropic earth models. For example, to model a TI formation, the five moduli associated with TI formations may be determined by a combination of measurements and assumptions. For example, the Sonic Scanner® downhole tool from Schlumberger may be used to measure three of the five moduli and two other may be determined by assumption.
Further, an ANNIE approximation may be used to approximate the two non-measured moduli. The ANNIE approximation is generally used in the seismic community to represent the behavior of laminated media (e.g., shale). The five parameters that are used to describe a TI material are C11, C13, C33, C44, and C66 where each Cij is an elastic coefficient in a stiffness matrix relating stress to strain of the subterranean formations. The parameter C12 is not an independent variable but may be determined by the following equation: C12=C11−2*C66 due to the symmetry of a TI formation. The Sonic Scanner® may be used to measure C33, C44, and C66. The two parameters C11 and C13 may be determined some other way, such as using the ANNIE approximation. Using the above measurements and approximations, an earth model for the TI formation may be generated.